As the use of electronic voting systems and e-commerce systems increases, the efficient batch verification of digital signatures becomes more and more important. In this paper, we first propose a new method to identify bad signatures in batches efficiently for the case when the batch contains one bad signature. The method can find out the bad signature using smaller number of modular multiplications than the existing method. We also propose an extension to the proposed method to find out two or more bad signatures in a batch instance. Experimental results show that our method yields better performance than the existing method in terms of the number of modular multiplications.

We propose a cooperative cognitive radio network (CCRN) with secondary users (SUs) employing two simultaneous transmit and receive (STAR) antennas. In the proposed framework of full-duplex (FD) multiple-input-multiple-output (MIMO) CCRN, the region of achievable rate is expanded via FD communication among SUs enabled by the STAR antennas adopted for the SUs. The link capacity of the proposed framework is analyzed theoretically. It is shown through numerical analysis that the proposed FD MIMO-CCRN framework can provide a considerable performance gain over the conventional frameworks of CCRN and MIMO-CCRN.